The Computable Differential Equation Lecture ... - Bruce E. Shapiro

108 CHAPTER 5. RUNGE-KUTTA METHODS
Hence the iteration formula (equation 5.155) gives
y n = y n−1 + hf(y n−1 ) + 1 2 h2 f y f + 1 6 h3 (f 2 y f + f yy f 2 ) + O(h 4 ) (5.162)
But since y ′ = f, y ′′ = f y f, and y ′′′ = f 2 y f 2 + f yy f 2 , the exact expansion is identical.
Thus the method is at least O(h 3 ).(In fact the error is precisely O(h 3 ) but this is
left as an exercise).
Thus in contrast to explicit methods, implicit methods may have order higher
than the number of stages; we shall see that they may, in fact have an order as high
as twice the number of stages. The price we pay is that the method is implicit –
some additional work, such as a Newton iteration – is required to implement the
method.
Figure 5.6: Region of absolute stability for the method of equation 5.153. The region
of absolute stability is the region outside the indicated curve.
Implicit methods can be absolutely stable in a large fraction of the negative real
plane, making them potentially good stiff solvers. For example, the method 5.153
when applied to the test equation gives
Solving the pair of equations for K 1 and K 2 gives
K 1 = y n−1 λ + h 4 (K 1 − K 2 )λ (5.163)
K 2 = y n−1 λ + h 12 (3K 1 + 5K 2 )λ (5.164)
(5.165)
6λ − 4hλ2
K 1 =
h 2 λ 2 − 4hλ + 6 y n−1 (5.166)
6λ
K 2 =
h 2 λ 2 − 4hλ + 6 y n−1 (5.167)
Math 582B, Spring 2007
California State University Northridge
c○2007, B.E.Shapiro
Last revised: May 23, 2007